A technique of the above kind is used in practice for modeling asymmetrical spaces or structures and the processes that take place in said spaces or structures. Examples of this are the simulation of the flow of heterogeneous media in apparatuses, the flow of air in a three-dimensional space surrounding objects, the calculation of electrical fields or the modeling of oil reserves and the flow of fluids through the subsurface of the earth.
Generally, a tetrahedral grid is used for a continuous three-dimensional space or structure, which grid is very flexible as regards its shape and which can be adapted to practically any space or structure. A drawback, however, is the arbitrary nature of said grid, which makes it necessary to store a large amount of information. Said arbitrary nature also leads to complicated algorithms for modeling the grid. As a result of the enormous amount of information and the resulting relatively long processing time by a digital processing unit, such as a computer or a processor, it is necessary in practice to limit the resolution of such a three-dimensional grid. As a result, discontinuities in the structure or space, i.e. discontinuous transitions in properties within the space or structure, such as material transitions caused by shifts in layers of the subsurface of the earth, cannot be modeled with sufficient accuracy and flexibility.
Accordingly, various kinds of regular multidimensional, generally three-dimensional, grids for modeling the subsurface of the earth, for example, have been developed in practice. The simplest grid is the so-called “voxel” model, which is a regular three-dimensional grid built up of cells having fixed dimensions and a fixed number of cells in each of the three directions. The cells are specified by three indices: I, J, K, one for each direction. It will be understood that this grid can only provide a very crude approximation of the complexity of the subsurface of the earth in which layers have shifted and folded or broken relative to each other as a result of the movement of the earth's crust. The advantage of this model is that such a grid can be defined with a limited number of parameters, as a result of which the amount of information to be processed, and consequently the computational time, are significantly reduced in comparison with the aforesaid flexible grid.
A better approximation of the actual geometry of the geological layers is possible by using an improved version of the simplified model, in which stacks of multidimensional cells are used and the stacks in question can be shifted relative to each other in a grid. In that case the geometry of the cells must be such that the cells fill the entire three-dimensional space as much as possible in a regular manner. Also when this improved model is used, problems arise when modeling discontinuities that extend at an angle through the geological layers, for example.
Further improvements of this grid model comprise the arranging of the stacks along a discontinuity, which leads to problems near the transitions of a discontinuity, however, for example in the case of geological layers that have shifted relative to each other along slanted fault lines. The fact is that cells having strongly deviating shapes and a relatively small volume, sometimes practically equal to zero, are formed at the transitions of the discontinuities as a result of stacks being collapsed. This leads to modeling artefacts, resulting in a distorted picture of the actual situation.
An example of such a technique is disclosed in U.S. Pat. No. 4,991,095, in which a stack of cells is adapted to a discontinuity, which leads to distorted or deformed cells or stacks of deformed cells, which makes it necessary to carry out complicated and time-consuming calculations and which leads to modeling errors and artefacts.
From U.S. Pat. No. 6,106,561 there is known a so-called “corner point grid” calculation, in which the cells or stacks of cells are aligned with discontinuities, which eventually leads to deformed cells as well, with the same drawbacks as described above.
In the article “One More Step in Gocad Stratigraphic Grid Generation: Taking into Account Faults and Pinchouts”, by Bennis, C. et al. in Proceedings of the NPF/SPE European 3-D Reservoir Modeling Conference, Stavanger, NORWAY, 16-17 Apr. 1996, pages 307-316, a technique is described wherein separate, closed spaces between discontinuities are defined, which spaces are individually provided with grids. Subsequently, the grids are coupled together on the discontinuities. Said coupling necessitates the introduction of artificial boundaries, which will likewise result in an irregular modeled structure.